The Formal Structure of Dialectical Psychology

Dialectical psychology (Riegel, 1973) postulates that one's
mental
processes move freely back and forth among all the Piagetian
stages,
meanwhile "transforming contradictory experience into momentary
stable
structures." In this paper it is shown that the symmetric
difference
operation of set-theoretic topology, together with its
complement, can be
used to represent the fundamental operations of both dialectical
logic and
dialectical psychology. Thesis and antithesis are expressed by
the
symmetric difference; synthesis and context, by its complement.
Applications of this algorithm are made to Piagetian
developmental
psychology, memory and learning, intelligence, quantitative
psychology,
creativity, and social psychology.

Introduction

"Dialectic" is a word of many meanings (Rychlak, 1976). Here the
meaning
of "dialectic" will be taken to be number (8) in the article
"Dialectic"
in the Encyclopedia of Philosophy: "...the logical development
of thought
or reality through thesis and antithesis to a synthesis of these
opposites." The distinction between an object or concept and
what it is
not leads inevitably to a dialectic view of the nature of the
world.
Think of figure and ground: everything has its contrast. Each
thought is
a composite consisting of some element (or elements) that belongs
to a
universal class (the thesis) coupled with what it is not within
that
universal (the antithesis) plus a means (the synthesis) of
resolving the
contradiction arising out of that discrimination.

Dialectical thinking consists of an exploration of contradictory
possibilities that results in cognitions which reduce cognitive
dissonance. Doubt searches out every belief: "It could be that
way, but
is it really?" To any appearance, there is the underlying
reality ("the
thing in itself"). Kahneman and Miller (1986) state that all
perceived
events are compared to counterfactual alternatives,
counterfactual in that
they constitute alternative realities to that experienced.
Johnson-Laird
(1995) in his study of mental models appropriate to deductive
thinking,
also notes the importance of counterfactuals. Knight and
Grabowecky
(1995) assert that counterfactuals are omnipresent in normal
human
cognition.

Curiosity, interest, and belief - "intentionality" - are inherent
in such
a dialectical structure though not in such a closed system as
logical
deduction. According to Marcel and Bisiach (1988),
"intention/al/ity" can
mean any one of the following:

A totality of objects or attributes comprehended as a
set
or concept. This is "intension."

An explicit goal or purpose - an "intention" - with
respect
to some particular thing.

The first of these - set-theoretic "intension" - is the meaning
employed
in this paper. Intentionality here consists of psychological
sets of
attributes, perepts, and/or concepts.

Riegel's postulates for dialectical psychology are the same as
Hegel's for
dialectic:

I. The unity and struggle of opposites.

II. The transformation of quantitative into qualitative
change.

III. The negation of the negation.

In the present study these three "laws" will be expressed, in the
contexts
of both dialectical logic and dialectical psychology, by means
of the
set-theoretic operation of symmetric difference: "one or the
other but not
both together." There exists a bewildering variety of notations
for the
symmetric difference, most of which have already been preempted
in other
contexts. Here I introduce a new one, $, which is like "S" for
"symmetric" and close to the set subtraction symbol \ as well;
besides it
is already on the keyboard.

Before discussing the dialectical "laws" in symbolic terms, it
seems
advisable at this stage to list the notations to be employed in
the
sequel:

/ ......... "factorization" of a set to yield a quotient
set, or, depending upon context, an arrow pointing upward to the
right

< ......... set containment symbol

0 ......... the null set

The symmetric difference operation $ upon two cognitive sets C
and C' is thus defined symbolically by the expression:

C $ C' = (C & -C') v (-C & C') =
C v C' \ (C & C')

(Equation 1)

Here $ denotes the symmetric difference of C and C'; & is the set
product
or intersection; v is the set-theoretic sum or union; -C denotes
the
set-theoretic complement of C, and similarly for -C'; and \ is
the
(relative) set-theoretic difference. A Venn diagram for $ looks
like that
in Figure 1.

Figure 1. : A Venn diagram for the symmetric difference of two
sets C and C'. The shaded portion comprises C $ C'. U is the
universe of discourse.

Exemplified in Eq. (1) and Figure 1 are the first two of the
Hegel-Riegel
"laws:" the unity and struggle of opposites and the replacement
of the
quantitative by qualitative, set-theoretic operations. Also
apparent is
the "law of the excluded middle." The symmetric difference and
its
complement are equivalent in logic to the Sheffer stroke, which
can of
itself generate the operations of Boolean algebra (Sheffer,
1913), and
so, ordinary first order logic. Formal "logical" thought is thus
accessible via the symmetric difference even though intuitive
thought is
more fundamental (Hoffman, 1980a,b; Riegel, 1973).

Synthesis of C and C', the generation of their commonality, comes
about by
taking the "negative of the negative." Here, however, the
innermost of
these two "negatives" is not the complement as such but rather
the
symmetric difference, as suggested by Eq. (1). The symmetric
difference
in Eq. (1) acts to strip C and C' of their commonality, C & C'.
If now
one takes the complement of (1), not only is the commonality C
& C' of C
and C' restored but also their context, "everything else" in U,
as in
Figure 2. The derivation goes as follows:

Figure 2. The "negation of the negation": not(C $ C') is the
set-theoretic union of the intersection of C and C'
with the intersection of notC and notC': -C & -C'.

It is postulated that Eq. (2) properly constitutes the
realization of the
third of the Hegel-Riegel "laws," the "negation of the negation,"
even
though the latter is traditionally taken to be simple
complementation.
The first term on the right in Eq. (2) represents the synthesis,
the
commonality of C and C'. This is Hegel's "unity of opposites."
The
second term provides their context within the universe of
discourse U.

So, via -(C & C'), we have not only a means of synthesis in this
second
phase but also divergent as well as convergent thinking in the
relation of
C and C' to "everything else," expressed by -C & -C' = -(C v C').
It is
Eq. (2) that corresponds to Hegel's principle that "negation is
determination" (Stace, 1955, p. 94).

Dialectical Logic

The foregoing has dealt with Fichte's famous triad:
thesis-antithesis-synthesis. Hegel's first triad is
"being-nothing-becoming." "Being" is positive; it affirms
existence. But
(Harris, 1987, p. 174):

... mere being, without discriminable character, is no
identifiable object. It has no content and is altogether
empty,
identifiable in effect with nothing.

Thus the symmetric difference of an object with itself is the
null set 0,
"nothing":

C $ C = (C & -C) v (-C & C) = (C & -C) = 0

(Equation 3)

A passage of "nothing" into "not-being" takes place via $ and
complementation in the comparison

C $ -0 =
(C & --0) v (-C & -0) =
0 v -C =
-C, notC

(Equation 4)

Next the symmetric difference of an object with "what it is not"
becomes -
is - "everything:"

C $ -C =
(C & C) v (-C & -C) =
C v -C =
U, the universe of discourse

(Equation 5)

By the same token, if C should happen to be contained within C',
C < C',
as in Figure 3, then C $ C' leads to the relative complement of
C in C'.
Here $ acts as differentia in Hegelian dialectic:

Figure 3. If C is a subset of C', then the symmetric
difference
of C and C' is the relative complement of C in C'.

Before proceeding on to the usages of the symmetric difference
in
Dialectical Psychology we note parenthetically that Hegel himself
would no
doubt have found such an algorithmic description of dialectic an
abomination. Styazhkin (1969, p. 112) has commented in
connection with
Hegel's denunciation of Ploucquet's "logical calculus" that "One
can only
imagine what epithets Hegel would have bestowed on contemporary
mathematical logic!" Yet Ploucquet's calculus for generating a
complete
description of all logical relations that was based only on an
identity
function and an inconsistency function seems rather close to
Hegel's own
ideas on synthesis and thesis-antithesis.

Application to Dialectical Psychology

In keeping with the formulation laid down in Eqs. (1) and (2),
it is
postulated (Hoffman, 1989, 1995) that cognitive processing is a
two-stage
phenomenon. The first phase consists of the action of the
symmetric
difference on either an ensemble of percepts and/or concepts or
the
contents of Working Memory (hereafter WM) in the style of Eq.
(1). The
second phase is postulated to follow as an integral part of the
process in
the form of the "negation" of phase one, as in Eq. (2). As
Ambrose Bierce
remarked, "The human brain not only can hold two contradictory
ideas at
the same time but insists on it."

In psychology the "similarity-difference" paradigm is of course
an old
story (Friedman, 1984, p. 113ff.). However, the paradigm should
read
"difference-similarity," for this order appears more appropriate
not only
in terms of the symmetric difference algorithm but also for
greater
realism. It is more important to a creature's survival in "the
jungle out
there" to be aware of novel stimuli in the environment and
properly
classify them than to reflectively seek their commonalities
(compare
Figure 2).

Piagetan Developmental Psychology

As Riegel viewed Piaget's four stages
of development, they were not mutually exclusive but rather
comprised a
system wherein an individual's thought processes can roam freely
back and
forth through all the Piagetian stages, as the situation may
demand. As
Riegel (1973, p. 367) put it,

... the option to operate simultaneously or in short
succession at different levels ... implies contradiction
andis dialectic in character.

One of the most compelling features of Riegel's theory was his
resolution
of the contretemps surrounding the fourth Piagetian stage of
development:
Formal Operations. The disconfirming experimental results of
Lovell
(1961), the finding that 37% of college students and 50% of
adults fail to
demonstrate Formal Operations thinking, as well as the prevalence
of such
grammatical errors as double negatives in everyday speech, all
cast doubt
upon "logical thought" as such. Rather, according to cognitive
research
(Bruner and Olver, 1963, p. 434), the mind acts to reduce the
number of
possible alternatives and select among these on the basis of
their
relative degrees of interest.

Riegel (1973, p. 354), using an approach based on McLaughlin's
"Psychologic" for the Piagetian stages, organized the latter in
terms of
the number of classificatins that a child is able to carry out
simultaneously at any given development period. During the
period of
Concrete Operations, the child is able to do double
classifications and
can form such logical constructs as C and C', C but not C', C'
but not C,
and neither C nor C' (note 1). Thus both the Piagetian period
of Formal
Operations and Riegel's dialectical operations stem from the
two-phase
cognitive operation based on the symmetric difference. Given the
foregoing logical constructs, both C $ C' and not(C $ C') are
implicit.

At the time that Formal Operations should be developing most
adolescents
seem to exhibit protean ways of thought more often based on value
judgments and peer pressure than logical reasoning. The latter
bear
little resemblance to the scientific method as envisioned by
Piaget. To
resolve contradictions, dialectic is required, not logic; logic
comes
afterward, if at all. This is not to say that one cannot do
logic, write
structured prose, compose music, solve puzzles, play chess, or
proceed in
logical thought processes when the occasion demands. But such
highly
formal thought processes invariably build upon the trains of
thought
previously generated intuitively in a sort of "chasing around the
cognitive diagram" (note 2). It is worthy of note in this
connection that
any novel - indeed any literary work with a plot - arouses the
reader's
interest largely because of the conflicts and contradictions
posed in that
plot and which the reader feels impelled to pursue further in
order to
resolve.

Memory

The distinction between the posterior perceptual systems of the
brain and the frontal-inferotemporal systems which mediate
semiotic
aspects and generate cognitive "plans" has been emphasized by
several
authors (Pribram, 1960; Hoffman, 1977, 1980a,b, 1985). Percepts,
which,
on the basis of veridicality, are the same in lower animals as
in humans,
are thus to be distinguished from cognitions, the meanings and
significances of things. Certainly, the semiotic aspects of
perceiving
the form of a mouse are different for a person's cat than for the
person
himself/herself.

I will thus identify Working Memory (WM, also denoted by W in the
subsequent analysis) with a small set of concepts or percepts,
the set
having been called up by directed thought from the Long Term
Memory Store
(hereafter LTM or L), that are presently in the forefront of
attention
(Baddeley, 1993). The elements of W interact with the Perceived
Field
(denoted hereafter by P = S/A or Q = imagery/A), where S is the
sensory
field, e.g., a non-attention-selected ensemble of forms that make
up the
visual field in its entirety. Thus S is "factored" by attention
A to
provide a subset P = S/A of the full perceptual manifold.

Many features of this triad of P, WM, and LTM seem well described
by the
symmetric difference algorithm. If the contents P of the
perceived field
have already been noted as W in WM, then either P < W or else P
= W. When
P < W, P $ W = W \ P. When, however, P and W are the same, then
P $ W =
0, and closure takes place. In either case identification occurs
within
WM and attention can be safely directed elsewhere. If, on the
other hand,
P is not contained within W, them P $ W leads to classification
and
discrimination based on the differences between WM and the
perceived field
P $ W.

Figure 4. The relations among P, WM, and LTM imparted by
the
symmetric difference. (a) Percepts or images already in WM.
(b) Novel percepts or images.

Further, in the case of assimilation when P < L, the commonality
of LTM
with the interaction between P and WM simply yields the latter
back again:

L & (W $ P) =
(L & W) $ (L & P) =
W $ P

(Equation 7)

since, as is evident, WM must be embedded in LTM, consciously or
subconsciously.

It is axiomatic that no one's memory contains all knowledge,
present and
future. The latter therefore constitutes the universe of
discourse,
denoted in the present context by K. The complement of LTM in
K
therefore represents "the great unknown." This more general
situation
occurs when P is not necessarily contained within LTM. Working
memory WM
is always contained within LTM, W < L, but if P is not contained
within L,
this may necessitate accomodation.

The symmetric difference among L, P, and W leads not only to the
commonality of P and W, but also "leads memory outside of itself"
in an
exploration of LTM for concepts outside of P and W. The argument
runs as
follows:

those
elements of memory outside of this combination - a reaching out
to
"everything else" within the Long Term Memory Store, and/or

a
combination of the perceived field and all knowledge K which lies
outside
of what the subject presently knows - an open-ended encounter
with "the
great unknown."

Thus the triple symmetric difference among P, W, and L
admits an expansion of what is presently known to new entities
and
concepts, in other words - accomodation.

Learning

One of the most successful university lecturers that I know
always makes it a point to incorporate counterexamples into his
lectures
as well as illustrative examples. Not only is the student's
interest
captured by the puzzle involved in the apparent paradox, but
he/she also
becomes aware of the boundaries of the subject: "what is" and
"what is
not," as well as its context. Research on educational learning
seems to
have taken a dialectical tack of late in the form of a new
emphasis on
removing contradictions from "naive theories" (Resnick, 1983).
The
concept-maps of Novak and Gowin (1984), which develop improved
understanding and retention through the resolution of
contradictions and
use of hierarchical classification, are also a case in point.
As Novak
and Gowin (1984, p. 19) put it:

Reflective thinking is controlled doing, involving a pushing
and pulling of concepts, putting them together and
separating
them again.

And also (ibid, p. 20),

Learning the meaning of a new piece of knowledge requires
dialog, exchange, sharing, and sometimes compromise.

Such processes seem very close to the two-phase thought processes
postulated in the symmetric difference model.

Intelligence

A commonplace that seems to run through the literature on
intelligence is that a reduction of confusion - a "quick study"
- rapid
insight - is an important element of the phenomenon. The first
phase of
the cognitive process in the symmetric difference model acts to
diminish
any confusing common attributes. All that remains after the
application
of $ to C and C' is the separate essentials of each, with no
overlap, no
"confusion." This separation process readily admits
discrimination and
classification and permits attention to be concentrated on the
truly
significant aspects in the second stage, the complementation of
the
symmetric difference, wherein the cognitive dissonance generated
by phase
one is resolved. In this second phase, whatever C and C' may
have in
common is registered - "convergent thinking."

The left side of the human brain (hereafter LHC) is supposed to
be devoted
mainly to mental processes of a sequential, analytical,
"logical," or
linguistic kind. The right brain (hereafter RHC), on the other
hand, is
supposed to process mainly psychological phenomena of a Gestalt
nature:
holistic and relational thinking that involves imagery,
imagination, and
spatial relations. RHC processing may thus be subsumed under the
broad
heading of "intuition." Of course such functions are principal
but not
exclusive functions of each hemisphere.

The situation is depicted in broad outline in Figure 5. The LHC
acts to

impose constraints upon the RHC via subcortical and commissural
interactions. Given the nature of mental processing, it would
appear that
a stimulus or other mental pattern is separated into distinctive
classifications in the LHC and processed for synthesis-Gestalt
aspects in
the RHC by a two-phase mental operation that has the character
of the
symmetric difference algorithm. Chen (1981) and Sinatra (1984)
have
extensively analyzed gifted intellect on the basis of such a
LHC-RHC
dichotomy and made a convincing argument for the importance of
such a
specialization. Zernhausern et al. (1981) have tested such a
hypothesis
and validated it for widely differing learning styles.

Quantitative Cognitions and Behavioral Decision Making

Dialectical
psychology postulates that quantitative change is transformed
into
qualitative change. Yet any psychological paradigm that does not
admit
quantitative and statistical phenomena would be lacking. The
bridge to
metric and probabilistic properties is provided in the present
instance by
the following theorem (Moran, 1968, p. 211): A family {C} of
sets on
which there is defined a measure m(C) can be made into a
quasi-metric
space by defining as a distance function d(C,C') between any two
members
C and C' the following expression:

d(C,C') = m(C $ C')

(Equation 9)

The symmetric difference thus provides a scale and dimensional
properties for such a family of sets. Such sets may correspond
in the
present context to any contents of P and/or WM, in particular the
standard psychometric measures.

Another important measure on sets is probability measure. A
probability
measure p(C) leads immediately, as a consequence of Eq. (1), to
the
formula

p(C $ C') = p(C v C') - p(C & C') =
p(C) + P(C') - 2 p(C & C')

(Equation 10)

while for the complementary second phase,

p[-(C $ C')] =
p(C & C') + p(-C & -C') =
1 + p(C & C') - p(C v C').

(Equation 11)

Formulas (10) and (11) appear relevant to the subjective
statistical
processes involved in behavioral decision making. The doubling
of the
product term in Eq. (10) provides a sharper basis for
classification and
discrimination than the standard probability formula for the
non-disjoint
union of events, and in Eq. (11) a trade-off between judgments
of
commonality and context is apparent. Eq. (11) admits an
immediate and
extensive connection with context and LTM via the term involving
notC and
notC'. These formulas also appear to embody such concepts as
Kahneman and
Tversky's (1982) "principle of complementarity" and "anchoring"
as intuitive
subjective probability rather than classical formal probability
theory.
This "representativeness' theme that intuitive probability is
context and
memory dependent rather than ordinary numerical runs through many
of
Kahneman and Tversky's conclusions. Such a view is in accord
with some
further basic properties of $:

p(C $ 0) = p(C) and p(C $ U) = p(C).

(Equation 12)

In these two results it is as if alternatives to the event C
simply did
not occur to the subject, a phenomenon frequently noted by
Kahneman and
Tversky.

Creativity

Cognitive conflict/dissonance leads to new accomodations and
new horizons. Such a passion to resolve discrepancies and
inconsistencies
is postulated here as the basis for creativity (Hoffman, 1995).
True
creativity demands not only problem solving and dialectical
insight but
also problem finding. An old paradigm is cast into the discard
and
replaced with one that has fewer contradictions and is more
satisfying
to the intellect. The creative act is based on a precursor
intuitive
exploration of the situation, demands a burning interest in the
problem,
and requires the sort of thinking that is implicit in RHC
imagery,
metaphor, and synthesis (Sinatra, ibid). The mental requirements
for this
process are traditionally listed as critical thinking,
independent thought
and judgment, self-starting, and perseverance, all of which
exhibit the
nature of dialectical thinking.

Social Psychology

In societal relationships, the cognitive role of
contradictory concepts becomes conflicting interests. Any
negotiation
process involves dialectical thinking. Negotiation proceeds from
certain
aspects upon which the parties are in accord to identification
and
comparison of the differences that separate them. If successful,
the
process culminates in a reconciliation of these differences by
a
commonality of interests.

Consider Riegel's (1977) simple dialogue between two speakers,
A and B, as
in Figure 6.

A --> A' --> A" --> ...
\ / \ / \ /
B --> B' --> B" --> ...

Figure 6. Riegel's simple dialogue between two speakers A
and B.

The primes and double primes denote successive times
A denotes the content of speaker A's initial statement, A', the
content of his statement on the second occasion, etc., and
similarly for
speaker B. Each speaker assimilates the other's statements: (A
$ B), etc.
and accomodates his own production -(A $ A') and -(B $ B') so as
to
elaborate and extend the preceding dialogue. The contexts of
these
operations can extend back in a set of paths to the full past
history of
the dialogue/negotiation. In such a simple dialectical process
as that in
Figure 6 we encounter the essence of social interaction.

Conclusion

In this paper an algorithmic structure for Dialectical Psychology
has been
investigated. The symmetric difference operation applied to
perceptual
and cognitive phenomena describes the key features of dialectic
and
dialectical reasoning. The complement of the symmetric
difference
completes the picture by providing "negation of the negation" as
well as
convergent and divergent thinking. Thus, as viewed in
Dialectical
Psychology, consciousness moves forward in time ceaselessly
engaged in a
stream of transformations, both psychological and physiological,
which
"satisfice" with respect to cognitive conflict.

The symmetric difference applied to two cognitive entities
generates the
standard recognition paradigm of discrimination and
classification -
"similarities and differences" - but in the opposite, more
realistic
order. A novel but more realistic interpretation of "negation"
in
classical Hegelian dialectic is then possible. Negation not of
negation
itself but of the symmetric difference leads to the second phase
of the
Dialectical Psychology process, which generates both synthesis
and
memory-cum-imagination. In the negation,

-(C & C') = (C & C') v (-C & -C')

(Equation 2, again)

the presence of -C & -C', i.e., notC and notC', makes possible
a more
general intuitionistic logic in which the law of the excluded
middle
expressed by Eq. (1) need no longer hold.

Notes

Note 1. The last of these is the Sheffer stroke, from which
all other
logical operations are derivable. See (Sheffer, 1913). [Editor's note: what is called the "Sheffer stroke" was actually originally derived by Charles
S. Peirce toward the end of the previous century; Sheffer rediscovered
it independently.]

Note 2. Riegel (1973, p. 363) put it this way,
It has never been shown convincingly that the highest level
of operation, i.e., Formal Operational intelligence,
characterizes
the thinking of mature adults. Only under the most
exceptional
conditions of logical argumentation and scholastic disputes
is it
conceivable that a person would engage in such a form of
thinking.
... Such forms of thinking merely provide the last straw in
the
process of scientific inquiry which is applied after
intuitive
thought is exhausted.